Geodesic
Isomorphic Group Rings of Free Abelian Groups
Canadian journal of mathematics, Tome 34 (1982) no. 1, pp. 8-16

Voir la notice de l'article provenant de la source Cambridge University Press

S. K. Sehgal ([9], Problem 26) proposed the following question : Let A, B be rings and X an infinite cyclic group. Does AX ⋍ BX imply A ⋍ B? M. M. Parmenter and S. K. Sehgal (cf. [9], Chapter 4) proved that, under some strong assumptions concerning rings A, B, the answer is affirmative. In this paper, we show that the assumptions concerning the ring B may be omitted in the above mentioned results. Moreover, it is proven that if (AX)X ⋍ BX then AX ⋍ B for all rings A, B. If A is commutative and noetherian then a partial answer to Problem 27, [9] follows from our results.Recently, L. Griinenfelder and M. M. Parmenter constructed nonisomorphic rings A, B for which the group rings AX, BX are isomorphic, [2], We give a new class of rings of this type. Our examples are noncommutative domains and algebras over finite fields. That also gives a negative answer to Problem 29, [9]. 
Krempa, Jan. Isomorphic Group Rings of Free Abelian Groups. Canadian journal of mathematics, Tome 34 (1982) no. 1, pp. 8-16. doi: 10.4153/CJM-1982-002-8
@article{10_4153_CJM_1982_002_8,
     author = {Krempa, Jan},
     title = {Isomorphic {Group} {Rings} of {Free} {Abelian} {Groups}},
     journal = {Canadian journal of mathematics},
     pages = {8--16},
     year = {1982},
     volume = {34},
     number = {1},
     doi = {10.4153/CJM-1982-002-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1982-002-8/}
}
TY  - JOUR
AU  - Krempa, Jan
TI  - Isomorphic Group Rings of Free Abelian Groups
JO  - Canadian journal of mathematics
PY  - 1982
SP  - 8
EP  - 16
VL  - 34
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1982-002-8/
DO  - 10.4153/CJM-1982-002-8
ID  - 10_4153_CJM_1982_002_8
ER  - 
%0 Journal Article
%A Krempa, Jan
%T Isomorphic Group Rings of Free Abelian Groups
%J Canadian journal of mathematics
%D 1982
%P 8-16
%V 34
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1982-002-8/
%R 10.4153/CJM-1982-002-8
%F 10_4153_CJM_1982_002_8

[1] 1. Coleman, D. B. and Enochs, E. E., Isomorphic polynomial rings, Proc. Amer. Math. Soc. 27 (1971), 247–252. Google Scholar

[2] 2. Griinenfelder, L. and Parmenter, M. M., Isomorphic group rings with non-isomorphic coefficient rings, Can. Math. Bull. Google Scholar

[3] 3. Lambek, J., Lectures on rings and modules (Blaisdell, 1966. Google Scholar

[4] 4. Lantz, D. C., R-automorphisms of R[G], G abelian torsion-free, Proc. Amer. Math. Soc. 61 (1976), 1–6. Google Scholar

[5] 5. Parmenter, M. M., Isomorphic group rings, Can. Math. Bull. 18 (1975), 567–576. Google Scholar

[6] 6. Parmenter, M. M., Coefficient rings of isomorphic group rings, Bol. Soc. Bras. Mat. 7 (1976), 59–63. Google Scholar

[7] 7. Parmenter, M. M. and Sehgal, S. K., Uniqueness of coefficient ring in some group rings, Can. Math. Bull. 16 (1973), 551–555. Google Scholar

[8] 8. Passman, D. S., The algebraic structure of group rings (John Wiley and Sons, New York, 1977). Google Scholar

[9] 9. Sehgal, S. K., Topics in group rings (Marcel Dekker, 1978. Google Scholar

Cité par Sources :